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Atwoods Machine Slides.pdf
- 1. Lab 4: Atwoodβs Machine PHYS 1003 Fall 2018
- 2. Atwoodβs Machine β’ Unbalanced weights on a pulley β’ One side accelerates up β’ The other side accelerates down β’ Magnitude of acceleration is the same π1 π2 Want to find π PHYS 1003 Fall 2018
- 3. Newtonβs Second Law β’ For linear motion β’ For angular motion Net Torque Moment of Inertia (sometimes called Rotational Inertia) Angular acceleration Atwoodβs Machine combines linear and angular motion ΰ· ΰ΄± πΉ = π ΰ΄± π ΰ· ΰ΄± π = πΌ ΰ΄± πΌ PHYS 1003 Fall 2018
- 4. Torques on the Pulley β’ Net torque (sum of all the torques present) π1 π2 π1 π2 π Frictional Torque (Capital Gamma) Linear acceleration Rotational acceleration ΰ· ΰ΄± π = πΌ ΰ΄± πΌ π2π β π1π β π€ = πΌπΌ π = ππΌ PHYS 1003 Fall 2018
- 5. use Divide by π Do some algebraβ¦. Now want to replace π2 β π1 π2π β π1π β π€ = πΌπΌ π2 β π1 β π€ π = πΌπΌ π π = ππΌ π2 β π1 = πΌ π2 π + π€ π PHYS 1003 Fall 2018
- 6. Forces on the Pulley β’ The net forces on the weights ΰ· ΰ΄± πΉ = π ΰ΄± π π2π β π2 = π2π π1π β π1 = π1(βπ) π1 β π1π = π1π π2 β π1 = π2 β π1 π β π1 + π2 π Mass 2 Mass 1 Combining the expressions for Mass 1 and Mass 2: PHYS 1003 Fall 2018 π1 π2 π1 π2 π
- 7. Now define the sum of the masses π And the difference of the masses Ξπ β¦ some more algebraβ¦. - from analysis of forces on the masses - from analysis of forces on the pulley π2 β π1 = πΌ π2 π + π€ π π2 β π1 = π2 β π1 π β π1 + π2 π Combining expressions: πΌ π2 π + π€ π = π2 β π1 π β π1 + π2 π π = (π1 + π2) Ξπ = π2 β π1 PHYS 1003 Fall 2018
- 8. β’ Isolate for π: β’ The masses are moving from rest in a straight line for a height change β From kinematics: β¦ some more algebraβ¦. π = Ξππ β π€ π π + πΌ π2 Ξπ¦ = π£ππ‘ + 1 2 ππ‘2 0 β = 1 2 ππ‘2 β PHYS 1003 Fall 2018
- 9. π ππ = π ππ π΄ + π° ππ βπ β π πππ π΄ + π° ππ π = Ξππ β π€ π π + πΌ π2 β = 1 2 ππ‘2 Combine* Rearrange* * Show how in your lab report PHYS 1003 Fall 2018
- 10. β’ Final expression to find π and π β’ Plot a graph of 1/π‘2 on the y-axis and π₯π on the x-axis Equation for a Straight line: π = ππ + π π ππ = π ππ π΄ + π° ππ βπ β π πππ π΄ + π° ππ πππππ = π ππ π΄ + π° ππ πππππππππ = β π πππ π΄+ π° ππ PHYS 1003 Fall 2018
- 11. β’ Need to measure: π, Ξπ, β and π β’ Need to collect a data set with 1/π‘2 and π₯π values for plotting β’ Want to find: π and π Need to measure π ππ = π ππ π΄ + π° ππ βπ β π πππ π΄ + π° ππ PHYS 1003 Fall 2018
- 13. Preliminary Measurements ππΉπ¬ = 2 Γ smallest division ππΉπ¬ = Β½ smallest division Mass measurements: β’ Mass of ten washes to find mass of 1 (mw) β’ Mass of m1 (mass with iron insert) and m2 Length measurement: β’ Distance travelled by m1 (β) Diameter measurement: β’ Diameter pulley to find the radius (π) ππΉπ¬ = 0.02 g PHYS 1003 Fall 2018
- 14. Table 1: Constant Parameters Masses Lengths Trial # Weight with the iron core+ screw Other weight + screw Mass of one washer (mass of all/10) distance travelled by π1 Radius of the pulley (diameter/2) π1 π2 ππ€ β π Units 1 2 3 Average Reading error ????? ????? Std Deviation (ΟπΌπ ) Std Dev mean (Οmean) Final Measurement (average Β± error) units What needs to be done with the error? PHYS 1003 Fall 2018
- 15. Reading Error vs Statistical Error β’ When have more than 1 measurement of the same quantity (eg. zero offset, mass, length, diameter) β’ Have to decide which error to use: reading error or statistical error ππΊπ« < πππΉπ¬ ππΊπ« β₯ πππΉπ¬ Use ππΉπ¬ Use πππππ ππΉπ¬ = reading error ππΊπ« = standard deviation πππππ = standard deviation of the mean π― = highest value in data set π³ =lowest value in data set π΅ = number of values in data set πππππ = ππΊπ« π΅ ππΊπ« = π― β π³ π΅ Inefficitent Statistics (when π΅ < ππ) PHYS 1003 Fall 2018 General Information, Section C
- 16. m1 m2 10 weights on π2 side m1 Move one weight to other side: Weight difference: Ξπ Setup m2 PHYS 1003 Fall 2018
- 17. m1 m2 β’ Reset button activates the solenoid and holds π1 in place β’ Start button causes it to drop and the timer to start β’ Timer stops when the mass hits the microswitch at the top β’ Collect 5 trials for each washer combination PHYS 1003 Fall 2018
- 18. The Clock Decimal Point Example: 0.26 s ππΉπ¬ = 0.02 s PHYS 1003 Fall 2018
- 19. Things to Measure (Logger Pro File) β’ Need to record 5 trials for each washer combination β’ Average drop time and itβs error should be generated for you (show sample calculations in your report) β’ Calculate mass difference PHYS 1003 Fall 2018
- 20. Mass difference βπ = π β 2π1 β 2πππ€ π = π1 + π2 + 10ππ€ ππ = ππ1 2 + ππ2 2 + 100πππ€ 2 Where: Must show 1 sample calculation in lab report Number of washers transferred PHYS 1003 Fall 2018
- 21. β’ The value for πΌ/π2 is given as 80 Β± 1 g β’ Obtain slope and intercept values (with errors) from the linear fit of the Logger Pro plot of 1/π‘2 vs π₯π β’ Use slope to determine the experimental value π Β± ππ β use kg and m, to get the answer in π/ππ β’ Use intercept to estimate the frictional torque π€ Β± ππ€ πππππ = π ππ π΄ + π° ππ πππππππππ = β π πππ π΄+ π° ππ PHYS 1003 Fall 2018
- 22. Compare Result with Theory Compare π you found experimentally (πππ₯π) with g from theory (ππ‘β) β= πππ₯π β ππ‘β πβ = ππππ₯π 2 + πππ‘β 2 β πβ β€ 2 Consistent πππ₯π Β± ππππ₯π ππ‘β Β± πππ‘β Assume πππ‘β = 0 PHYS 1003 Fall 2018 General Information, Section C
- 23. Before You Leave β’ All measurements: masses (system, washers), diameter of pulley, height of drop, 5 sets of 5 drop times β’ Your printed graph of 1 π‘2 π£π βπ β’ Make sure you know what to do to find π and π€ PHYS 1003 Fall 2018
- 24. β’ Report due at the end of the lab β’ MUST come prepared! β’ Do Purpose, Theory, Apparatus at home β’ Pre-research Discussion topics Next Lab: IN-CLASS!!!